Its going to encourage a lot of moisture to bead up in that area when taking a hot shower, especially if ventilation is lacking in anyway. You may get mildew/mold in the cabinet over time if there is going to be no insulation at all due to the depth of the cabinet. If you have a bit of room for insulation behind it I suggest using form fitting, cut foam board with the highest R-value you can get. Foam board can have an R value up to about 6.6 per inch, almost double that of fiberglass, so even just an inch of insulation is going to be about as good as 2 inches of fiberglass batt. I'd use the foam board and then use some spray foam around it in any little nooks or cracks. Better yet if there is room use foam board with a radiative barrier (facing in toward the heated space), and leave a small air gap if it'll fit, say 1" foam board with a 1/2" air gap if there is room will likely give you around an R value of 10, about what that area of wall would have normally (a little less).

Otherwise if there is no insulation you are going to have less insulative value then a double pane window (maybe as much as a leaky double pane window) in that area.

Make sure that if you do this to properly frame out the area as, just like a window or door, you're going to need the proper framing, cripple studs, king studs, header beam, etc. Otherwise you are removing part of your load bearing wall and leaving it unsupported.

I am remodeling a bath--would like an inset cabinet in an outside wall. Would the lack of insulating this small space be an issue? Will the toothpaste be thick?

A recommendation --- don't.

Even if you were to properly seal the opening with a vapour barrier ---- then add insulation around and behind the cabinet ... which will reduce the cabinet depth and width --- you end up with a narrower and a very shallow cabinet.
You have to ask yourself ----with all the potential problems with cold infiltration and moisture --- ending up with a cabinet that may only fit your toothpaste --- is it worth it ?